Mathematical economics

Mathematical economics is the use of math to understand and solve problems in economics. It helps economists create clear and exact descriptions of how money, resources, and choices work together. By using math, economists can test ideas and make predictions about complex subjects that are hard to explain with words alone.

Math lets economists make specific and testable statements about topics that might be debated. Many economic theories today are shown as mathematical models, which are simplified versions of real situations. These models help clarify what is being assumed and what results are expected.

Mathematical economics includes many kinds of problems, like finding the best way for a household or business to reach its goals, studying how markets stay balanced, and looking at how economies change over time. The use of math in economics began in the 1800s and grew a lot during the 20th century, especially after the Second World War, with the development of game theory. Some famous economists, like John Maynard Keynes, have questioned whether all human behavior can truly be described with math.

History

Main article: History of economic thought

The use of mathematics to study economics started in the 17th century. Early scholars in Germany focused on collecting data to help with public administration. In England, scholars used numbers to study government issues, a method called Political Arithmetick.

In the 19th century, economics began using more math. Scholars started creating models to describe economic behavior. One important model was created by Johann Heinrich von Thünen in 1826, who studied how farmland is used. Later, scholars trained in math from the physical sciences began applying these methods to economics.

Marginalists and the roots of neoclassical economics

Main article: Marginalism

Scholars like Augustin Cournot and Léon Walras used math to describe how people make choices to get the most benefit. Cournot studied competition between two sellers, while Walras looked at the entire economy. These ideas helped shape modern economics.

Modern mathematical economics

From the late 1930s, new mathematical tools like differential calculus and differential equations were used to improve economic theory. This was similar to how math was applied to physics before. These tools helped economists describe and understand economic behavior more clearly.

Vilfredo Pareto studied microeconomics by looking at how people make decisions to get what they want most. Paul Samuelson used math to find common patterns in economics in his book "Foundations of Economic Analysis" in 1947. He showed that economic systems can be modeled like other systems in nature.

John von Neumann created models of expanding economies using math. Wassily Leontief developed input-output analysis to show how changes in one part of the economy affect other parts. Mathematical optimization helps economists find the best solutions to problems, like how consumers and firms make decisions to get the most value. Game theory, developed by John von Neumann and others, studies how people and groups make decisions when they interact. Agent-based computational economics uses computer simulations to study complex economic systems made up of many interacting parts.

Mathematicization of economics

During the 20th century, most economics articles in important journals were written by university economists. These articles often focused on economic theory, which became more abstract and mathematical over time. In 1892, most articles did not use complex math, but by 1990, very few articles were without mathematical ideas. By 2003 and 2004, almost all articles in top journals used math or statistics to explain their ideas.

Econometrics

Main article: Econometrics

Ragnar Frisch coined the term "econometrics" and helped start the Econometric Society and the journal Econometrica. His student, Trygve Haavelmo, showed that careful statistical analysis could test economic theories using real-world data.

Henry L. Moore, an American economist, studied farm productivity and tried to match crop yields to mathematical curves. Though his methods had some flaws, his work helped develop economic modeling. Later, Nicholas Kaldor improved on Moore’s ideas with a model explaining business cycles.

Application

Mathematical economics uses advanced math, like calculus and matrix algebra, to study economic ideas. These tools help make complex economic problems easier to understand and solve. Many economic theories are best explained using math because they involve many variables and relationships.

Economists today rely heavily on math, and students studying economics usually need strong math skills. This has led to many mathematicians working in economics, using their knowledge to solve real-world economic problems. Economic models can be either stochastic, meaning they include random events, or deterministic, meaning they follow fixed rules.

Discussions of validity

Different groups of economists have had different opinions about using math in economics. Some, like the Austrian school, think that economics can't always be expressed with numbers and math. They feel it makes economics seem more like a science than it really is.

Other economists, like Paul Samuelson and Robert M. Solow, believe that math is very important in economics. They say it helps to explain complex ideas clearly and has led to important progress in understanding economics.